Question Video: Calculating the Distance between a Spacecraft and the Moon | Nagwa Question Video: Calculating the Distance between a Spacecraft and the Moon | Nagwa

# Question Video: Calculating the Distance between a Spacecraft and the Moon Physics

A spacecraft heading toward the Moon to deploy a rover on the surface has a mass of 870 kg. The magnitude of the gravitational potential energy of the spacecraft-and-Moon system is 427 MJ. How far away from the Moon’s center of mass is the spacecraft? Use a value of 7.35 × 10²² kg for the mass of the Moon and 6.67 × 10⁻¹¹ m³/kg.s² for the universal gravitational constant. Give your answer to 3 significant figures.

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### Video Transcript

A spacecraft heading toward the Moon to deploy a rover on the surface has a mass of 870 kilograms. The magnitude of the gravitational potential energy of the spacecraft-and-Moon system is 427 megajoules. How far away from the Moon’s center of mass is the spacecraft? Use a value of 7.35 times 10 to the 22nd kilograms for the mass of the Moon and 6.67 times 10 to the negative 11th cubic meters per kilogram second squared for the universal gravitational constant. Give your answer to three significant figures.

If we say that this pink circle is the Moon and that this is our spacecraft heading toward it, our problem statement tells us the mass of the spacecraft, let’s call that 𝑚 sub s; the mass of the moon, we’ll call that 𝑚 sub m. And it also tells us the gravitational potential energy of this system of two masses, the spacecraft and the Moon. We’ll call that magnitude 𝐺𝑃𝐸. Knowing all this, what we want to solve for is the distance between the spacecraft and the center of the Moon. And we can call that distance 𝑟.

To get started, we can recall the equation for gravitational potential energy in a radial gravitational field. This is the kind of field created by a spherically symmetric object, such as the Moon. For this kind of field, the gravitational potential energy shared between two masses, the larger one creating the field and the smaller one experiencing it, is equal to negative 𝐺, the universal gravitational constant, times those masses divided by the distance between their centers. In our scenario, it’s not 𝐺𝑃𝐸 we want to solve for though, but rather the distance 𝑟.

If we multiply both sides of this equation by the fraction 𝑟 divided by 𝐺𝑃𝐸, then on the left-hand side, that gravitational potential energy cancels out. And on the right, the distance 𝑟 cancels. And that leaves us with this equation. 𝑟 is equal to negative big 𝐺 times big 𝑀 times little 𝑚 divided by 𝐺𝑃𝐸.

One important thing to remember here is that this term, the gravitational potential energy in this equation, is negative. And so if we were to plug in actual values in this equation, we would have a negative multiplied by a negative, which is a positive, giving us then an overall positive value for 𝑟. We can apply this equation to our scenario like this. The distance between the center of the Moon and the spacecraft is equal to the universal gravitational constant times the mass of the Moon times the mass of the spacecraft all divided by the magnitude of the gravitational potential energy of this two-mass system. And notice that we’re given all of the values that appear on the right-hand side of this equation.

Our next step then is to substitute them in. So here we have the value for the universal gravitational constant. Here’s the mass of the Moon, here’s the mass of our spacecraft, and here’s the magnitude of the gravitational potential energy between these masses. As a last step, before we calculate 𝑟, we’ll want to convert this gravitational potential energy from units of megajoules into units of joules. That way, all the units in this expression will be on the same footing. Now, one megajoule is equal to 10 to the sixth or a million joules. So we can rewrite our denominator as 427 times 10 to the sixth joules. This number isn’t in scientific notation, but it is the correct number of joules of energy. When we calculate 𝑟, to three significant figures, we get a result of 9.99 times 10 to the sixth meters. We can equivalently express this as 9,990 kilometers. That’s the distance from the center of mass of the Moon to the spacecraft.

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